We develop a formalism for relative Gromov–Witten invariants following
Li J. Li, Stable morphisms to singular schemes and relative stable
morphisms, J. Differential Geom. 57 (3) (2001), 509–578, J. Li, A degeneration
formula of GW-invariants, J. Differential Geom. 60 (2) (2002),
199–293 that is analogous to the symplectic field theory (SFT) of
Eliashberg, Givental and Hofer Y. Eliashberg, A. Givental and H. Hofer,
Introduction to symplectic field theory, Geom. Funct. Anal. (Special Volume,
Part II) (2000), 560–673 GAFA 2000 (Tel Aviv, 1999). This formalism
allows us to express natural degeneration formulae in terms
of generating functions and re-derive the formulae of Caporaso–Harris
L. Caporaso and J. Harris, Counting plane curves of any genus, Invent.
Math. 131 (2) (1998), 345–392, Ran Z. Ran, Enumerative geometry of
singular plane curves, Invent. Math. 97 (3) (1989), 447–465, and Vakil
R. Vakil, The enumerative geometry of rational and elliptic curves in
projective space, J. Reine Angew. Math. 529 (2000), 101–153 for counting
rational curves. In addition, our framework gives a homology theory
analogous to SFT homology.